Question

Evaluate :

\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]

Answer 1

\[∆ = \begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]

\[ = a( a^2 - bc) - b(ca - b^2 ) + c( c^2 - ba)\]

\[ = a^3 - abc - bca + b^3 + c^3 - abc\]

\[ = a^3 + b^3 + c^3 - 3abc = (a + b + c)( a^2 + b^2 + c^2 - ab - bc - ca)\]